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Exceptional Lines and Excitation of (Nearly) Double-Pole Quasinormal Modes: A Semi-Analytic Study in the Nariai Black Hole

We show that quasinormal modes (QNMs) of a massive scalar field in Kerr-de Sitter and Myers-Perry black holes exhibit an exceptional line (EL), which is a continuous set of exceptional points (EPs) in parameter space, at which two QNM frequencies and their associated solutions coincide. We find that the EL appears in the parameter space spanned by the scalar mass and the black hole spin parameter, and also in the Nariai limit, i.e., $r_{\rm c} - r_{\rm h} \to 0$, where $r_{\rm c}$ and $r_{\rm h}$ denote the radii of the cosmological and black hole horizons, respectively. We analytically study the amplitudes or excitation factors of QNMs near the EL. Such an analytic treatment becomes possible since, in the Nariai limit, the perturbation equation reduces to a wave equation with the Pöschl-Teller (PT) potential. We discuss the destructive excitation of QNMs and the stability of the ringdown near and at the EL. The transient linear growth of QNMs -- a characteristic excitation pattern near an EP or EL -- together with the conditions under which this linear growth dominates the early ringdown, is also studied analytically. Our conditions apply to a broad class of systems that involve the excitation of (nearly) double-pole QNMs.